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Mathematical Treasure: Early Catalan Arithmetic

Frank J. Swetz (The Pennsylvania State University)

Suma de la art de Arismetica (Compendium of the art of arithmetic, 1482) is the first printed arithmetic book on the Iberian Peninsula and therefore the first printed arithmetic book in Portugal or Spain. It is written in Catalan. Little is known about the author, Francesc Santcliment, except that his life spanned the 15th and 16th centuries. The book is a practica written for the needs of merchants. The work includes economic and trade information, such as customs data on transactions involving pepper, saffron, cinnamon, honey, sugar, silk, and sheep, and details on weights, measures, and coins. The first page of text outlines the scope of the work. (The title of the book is in the third and fourth lines of text.)

Hindu-Arabic (or Indo-Arabic) numerals are introduced on the second and third pages of text (folios 1v and 2r) and are used throughout the demonstrations and explanations. The decimal nature of the numbers is stressed.

Multiplication is introduced as repeated addition. Below (ff. 18v-19r), rather interesting “Table of Multiplication Facts” is produced for the reader. Can you understand how it works?

In the pages below (ff. 37v-38r), the algorithm for the multiplication of two multi-digit numbers is familiar to a modern reader. Note the didactical aid of the use of inclined lines indicating the place-value indexing of partial products.

Below (ff. 34v-35r), division is performed by the “galley method.” The example given is \(5732894\div 457,\) but only three partial divisions are undertaken resulting in the first three digits, \(125,\) of the quotient, \(12544\frac{3}{5}.\)

On the pages below (ff. 44v-45r), a monetary exchange problem is encountered involving ducats, diners, and sous, units of currency foreign to the Spanish economy. The solution of the problem involves multiplication, division, and a use of fractions.

An application of “the rule of three” is demonstrated (ff. 46v-47r):

The reduction of common fractions employing the least common denominator is discussed (ff. 57v-58r):

These images are obtained through the courtesy of the World Digital Library where a complete viewing of the text is possible. The book itself is held by the National Library of Catalonia (Biblioteca de Catalunya), Spain.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Early Catalan Arithmetic," Convergence (June 2017)